I think this argument completely fails. The child has a natural right to eat. You don't get to put constraints on natural rights, especially those that are basic to human existence, especially based on a speculative unrealized risk of harm to others
Straight up at all levels of law and logic, an unrealized harm that has not occurred, may not ever occur and has no well known history of occurring is not ever going to be a consideration when balancing freedoms and rights.
As for your suggestions regarding alternative feeding options. Yeah, they work for some infants, sometimes and in some situations. But not for all infants, all times or all situations. And you are also ignoring the basic principle that neither third parties or government get to dictate basic parenting activities to parents.
Also you can't cover a boob without the cloth getting in or over the face of the child. When people speak of "covering" they generally mean draping a blanket or burp cloth over the mother's shoulder down to the child's shoulder covering both the breast and the child's head. This is frequently done not for privacy but to reduce outside stimulation in order to relax and soothe the child to sleep. Older infants who have higher activity to sleep ratios often refuse coverage because they are only hungry, not hungry and tired.
In this specific story, the mom in question had a tank style suit with a slit she used for feeding. This is a perfect design because the straps remain in place on your shoulders and provide maximum coverage. This is why I question whether the claims of the employees are true or if they simply made stuff up because of their own discomfort.
This happens a lot, which is why most if not all states have laws protecting public breastfeeding. Simply put, an infant's comfort is more important than everyone else's. That standard is as old as the hills because babies are helpless and needy and adults aren't.
You can grouse about the situation until the cows come home, but it's not change it one iota. Which is why the pool staff is getting re-educated, they were wrong, they violated the law and created an unnecessary scene while inconveniencing a family unit due to their own ignorance and preconceived discomfort.